A13935. Divide and Sum

You are given an array $a$ of length $2n$ . Consider a partition of array $a$ into two subsequences $p$ and $q$ of length $n$ each (each element of array $a$ should be in exactly one subsequence: either in $p$ or in $q$ ).

Let's sort $p$ in non-decreasing order, and $q$ in non-increasing order, we can denote the sorted versions by $x$ and $y$ , respectively. Then the cost of a partition is defined as $f(p, q) = \sum_{i = 1}^n |x_i - y_i|$ .

Find the sum of $f(p, q)$ over all correct partitions of array $a$ . Since the answer might be too big, print its remainder modulo $998244353$ .

The first line contains a single integer $n$ ( $1 \leq n \leq 150\,000$ ).

The second line contains $2n$ integers $a_1, a_2, \ldots, a_{2n}$ ( $1 \leq a_i \leq 10^9$ ) — elements of array $a$ .

Print one integer — the answer to the problem, modulo $998244353$ .

Two partitions of an array are considered different if the sets of indices of elements included in the subsequence $p$ are different.

In the first example, there are two correct partitions of the array $a$ :

1. $p = [1]$ , $q = [4]$ , then $x = [1]$ , $y = [4]$ , $f(p, q) = |1 - 4| = 3$ ;
2. $p = [4]$ , $q = [1]$ , then $x = [4]$ , $y = [1]$ , $f(p, q) = |4 - 1| = 3$ .

In the second example, there are six valid partitions of the array $a$ :

1. $p = [2, 1]$ , $q = [2, 1]$ (elements with indices $1$ and $2$ in the original array are selected in the subsequence $p$ );
2. $p = [2, 2]$ , $q = [1, 1]$ ;
3. $p = [2, 1]$ , $q = [1, 2]$ (elements with indices $1$ and $4$ are selected in the subsequence $p$ );
4. $p = [1, 2]$ , $q = [2, 1]$ ;
5. $p = [1, 1]$ , $q = [2, 2]$ ;
6. $p = [2, 1]$ , $q = [2, 1]$ (elements with indices $3$ and $4$ are selected in the subsequence $p$ ).